Original Citation:
Polynomial Meshes: Computation and Approximation
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in Science and Engineering, CMMSE 2015 6–10 July, 2015.
Polynomial Meshes:
Computation and Approximation
S. De Marchi1, F. Piazzon1, A. Sommariva1 and M. Vianello1
1 Department of Mathematics, University of Padova
emails: demarchi,fpiazzon,alvise,[email protected]
Abstract
We present the software package WAM, written in Matlab, that generates Weakly Admissible Meshes and Discrete Extremal Sets of Fekete and Leja type, for 2d and 3d polynomial least squares and interpolation on compact sets with various geometries. Possible applications range from data fitting to high-order methods for PDEs.
Key words: Multivariate Polynomial Approximation, Weakly Admissible Meshes, Discrete Extremal Sets.
MSC 2000: 41A10, 65D32.
1
Polynomial meshes
In the field of multivariate polynomial approximation, the notion of polynomial mesh has recently emerged as a significant concept. Originally introduced in the seminal paper [6], it has been studied in several subsequent papers, from both the theoretical and the computa-tional point of view; cf., e.g., [1, 2, 3, 5, 9, 11, 13, 15, 18, 19, 20, 22, 24, 25, 26]. Moreover, approximate Fekete-like points extracted from polynomial meshes have begun to play a role in the framework of high-order methods for PDEs, cf., e.g., [7, 28].
We recall that a polynomial Weakly Admissible Mesh (WAM) is a sequence of dis-crete subsets of a polynomial determining (polynomial vanishing there vanish everywhere) compact setK⊂Rd(or more generallyK ⊂
Cd) such that the polynomial inequality kpkK ≤C(An)kpkAn , ∀p∈P
d
n (1)
is satisfied, where both card(An)≥dim(Pdn) = n+d
d
and C(An) grow at most like a power
exceedingn, andkfkX the sup-norm of a functionf bounded on the (discrete or continuous)
setX. The quantityC(An) is often called “constant” of the WAM. WhenC(An) is bounded
we speak of an Admissible Mesh (AM), which is termed “optimal” if card(An) =O(nd). Among their properties, it is worth recalling the following ones (cf. [6]), which give also recipes to construct new from known WAMs:
• any affine transformation of a WAM is still a WAM,C(An) being invariant;
• any sequence of unisolvent interpolation sets whose Lebesgue constant Λn grows at
most polynomially withn is a WAM, with constantC(An) = Λn;
• afinite productof WAMs is a WAM on the corresponding product of compacts,C(An) being the product of the corresponding constants;
• a finite unionof WAMs is a WAM on the corresponding union of compacts, C(An)
being the maximum of the corresponding constants.
A special constructive role is played by some univariate interpolation sets, namely the Chebyshev-Lobatto nodes of an interval [a, b] (via the affine transformation τ(s) =
b−a
2 s+
b+a
2 ),
Xn(a, b) ={τ(ξj)} ⊂[a, b], ξj = cos (jπ/n), 0≤j≤n , (2)
the classical Chebyshev nodes
Zn(a, b) ={τ(ηj)} ⊂(a, b), ηj = cos (2j+ 1)π 2(n+ 1) , 0≤j≤n , (3) and the Chebyshev-like “subperiodic” angular nodes
Θn(α, β) =φω(Z2n(−1,1)) +
α+β
2 ⊂(α, β), ω=
β−α
2 ≤π , (4)
obtained by the nonlinear tranformation φω(s) = 2 arcsin sin ω2
s, s ∈ [−1,1]. These nodal sets satisfy the following fundamental inequalities (the first and second are well-known results of polynomial interpolation theory, the third has been recently proved in the framework of trigonometric interpolation, cf. [8] and the references therein).
Lemma 1 Let p ∈ P1
n be a univariate algebraic polynomial, and Xn, Zn the Chebyshev
nodal sets (2), (3). Let be t ∈ T1
n be a univariate trigonometric polynomial, and Θn the
angular nodal set (4). Then the following inequalities hold
kpk[a,b]≤cnkpkXn , kpk[a,b]≤cnkpkZn , ktk[α,β]≤c2nktkΘn , cn= 1 +
2
π log(n+ 1). (5)
Below, we recall some recent results on WAMs inR2andR3, privileging the constructive approaches that are at the base of the software package [10] (the case of surface meshes, in particular on algebraic surfaces, is under development). In all the constructions, Zn can
1.1 Planar meshes
1.1.1 Polygons
Anyconvexquadrangle with verticesv1,v2,v3,v4, is the image of abilinear transformation of the square, namelyx=σ(s1, s2) = 14 ((1−s1)(1−s2)v1+ (1 +s1)(1−s2)v2+ (1 +s1)
(1 +s2)v3+ (1−s1)(1 +s2)v4), where (s1, s2)∈[−1,1]2, with a triangle, e.g. v3 =v4, as
a special degenerate case. Using the fact thatp◦σ ∈P1
n⊗P1n for everyp∈P2n, by Lemma
1 we can easily prove the following
Proposition 1 The sequence of “oblique” Chebyshev gridsAn=σ(Xn(−1,1)×Xn(−1,1))
is a WAM of the convex quadrangleQ=σ([−1,1]2), with constantC(An) =c2n=O(log2n)
and card(An)≤(n+ 1)2.
Concerning general polygons, in view of a well-known result of computational geometry and the finite union property of WAMs, we have then immediately the following
Proposition 2 Let K be a simple polygon (convex or concave) with ` sides. Then K has
a WAM given by the union of the WAMs of`−2triangles of a minimal triangulation, with constant C(An) =c2n=O(log2n) and card(A
n)∼(`−2)n2.
The geometric constructions of Proposition 1 and 2, studied in [5, 13], are implemented by the Matlab functionswamquadrangle and wampolygon of [10] (where minimal polygon triangulation is generated by a Matlab version of the “Ear Clipping” algorithm).
1.1.2 Circular sections
Several circular sections, such as circular sectors, segments, zones and lenses, can be described by linear blending of arcs, cf. [25]. Let u(θ) = a1cos(θ) + b1sin(θ) +c1, v(θ) = a2cos(θ) +b2sin(θ) +c2, θ ∈ [α, β], be two trigonometric planar curves of de-gree one,ai= (ai1, ai2),bi = (bi1, bi2),ci = (ci1, ci2), i= 1,2, being suitable bidimensional
vectors (with ai,bi not all zero), with the important property that the curves are both
parametrized on the same angular interval [α, β], 0< β−α ≤2π. Consider the blending transformation x= σ(s, θ) = su(θ) + (1−s)v(θ), (s, θ) ∈ [0,1]×[α, β]}. Now, for every
p∈P2
n we have thatp◦σ∈P1n⊗T1n, and we can prove the following
Proposition 3 The sequence of “blended” grids An =σ(Xn(0,1)×Θn(α, β)) is a WAM
of K=σ([0,1]×[α, β]), withC(An) =cnc2n=O(log2n) andcard(An)≤(n+ 1)(2n+ 1).
Relevant arc related domains that do not fall in the previous class are circular lunes (difference of two overlapping disks). A lune, whose boundary is given by two circular arcs, a longer one with semiangle sayω2, a shorter one with semiangle sayω1, can be described by
different bilinear trigonometric transformations of rectangles (in angular variables), of the formx=σ(φ, θ) =a1+a2cos(θ)+a3sin(θ)+a4cos(φ)+a5cos(φ) cos(θ)+a6cos(φ) sin(θ)+
a7sin(φ) sin(θ), (θ, φ) ∈ R = [−ω1, ω1]×[−ω2, ω2], where the ai = (ai,1, ai,2) are suitable
2-dimensional vectors depending on ω1 and ω2, i.e., each component of σ is in the
trigono-metric space T11⊗T11; cf. [25] and the references therein. Hence for every p∈ P2n we have
that p◦σ ∈T1
n⊗T1n, and by Lemma 1 one can prove the following
Proposition 4 The sequence of curvilinear gridsAn= σ(Θn(−ω1, ω1)×Θn(−ω2, ω2)) is
a WAM of the lune K =σ(R), withC(An) =c22n=O(log2n) and card(An)≤(2n+ 1)2.
WAMs on arc blending domains and on circular lunes can be computed by the Matlab functions wamblendand wamluneof [10].
1.1.3 Convex and starlike C2-domains
In [20], a constructive approach has been studied for the generation of optimal Admissible Meshes on smooth starlike planar domains, based on the fulfillment of a tangential-like Markov polynomial inequality.
Proposition 5 Let K⊂R2 be a planar compact starlike domain. Assume that K satisfies
a Uniform Interior Ball Condition (UIBC), i.e., every point of∂K belongs to the boundary of a disk with radiusρ >0, contained in K (geometrically, there is a fixed disk that can roll along the boundary remaining inside K). Then, for every fixed α∈(0,1/√2),K possesses an optimal admissible mesh {An} such that C(An)≡
√
2
1−α√2, card(An)∼n
2 length(∂K)
αρ .
When K is convex with C2 boundary, ρ can be easily computed, being the minimal radius of curvature of the boundary, by the well-known “Rolling Ball Theorem”. The con-struction of optimal polynomial meshes on convex compact sets, defined by a level set of a smooth convex bivariate function, is implemented by the Matlab functionconvomesh [10]. 1.2 Solid meshes
1.2.1 Cones, pyramids, cylinders and solids of rotation
The following geometric constructions have been studied in [11], and implemented by the Matlab functionswamcone andwamrotof [10]. In both the propositions below, Ω⊂R3 is a
planar compact set where a 2-dimensional WAM, sayAn, is known.
Proposition 6 Letv∗ be a point inR3 not belonging to the plane ofΩ. Then the generalized
cone C with base Ω and vertex v∗, has a WAM, say Bn, with C(Bn) = cnC(An) and
segments joiningv∗ with each of the points ofAn. If we consider the truncated cone obtained
by cutting the cone with a plane parallel to the base, then the WAMBnis given by the union
of the Chebyshev-Lobatto points of the cut segments, and card(Bn) = (n+ 1)card(An).
Proposition 7 Let α be a line in R3 lying on the plane of Ω and not intersecting Ω (or
intersecting Ω only at the boundary), and let θ∗ ∈ (0,2π] a given angle. Then the solid of rotation R obtained by rotating Ω around the axis α by an angle θ∗ has a WAM, say Bn,
withC(Bn) =c2nC(An)andcard(Bn) = (2n+ 1)card(An), given by the union of the2n+ 1
copies of An corresponding to rotating An by the anglesΘn(0, θ∗).
The proofs resort to Lemma 1, together with the relevant affine transformations (in particular, to the classical Thales intercept theorem for the generalized cone case). Ob-serve that generalized pyramids(and thus tetrahedra) are special cases of cones, Ω being a polygon. By tetrahedralization and finite union, any simple polyhedron has a WAM with constantO(log3n) and cardinalityO(n3). Moreover,generalized cylinders(up to a rotation
compact sets of the form Ω×[a, b]) by the product property have a WAMBn=An×Xn(a, b)
with the same constant and cardinality of that of a truncated cone. A different WAM for a standard cylinder with circular base can be generated by rotation of the Padua interpolation points of a rectangle [9].
1.2.2 Lissajous meshes on the 3-cube
By the product property, (Xn(−1,1))3 is a natural grid-type WAM for the reference cube
[−1,1]3, with constant c3n=O(log3n). Recently, a new approach for trivariate polynomial
approximation has been explored, based on curves instead of grids. Indeed, it has been proved that Z [−1,1]3 p(x)p dx (1−x21)(1−x22)(1−x23) =π 2 Z π 0 p(`n(θ))dθ , ∀p∈P32n,
where`n(θ) = (cos(αnθ),cos(βnθ),cos(γnθ)), θ∈[0, π], is the Lissajous curve with integer
frequency parameters (αn, βn, γn) = 34n2+ 12n,34n2+n,34n2+32n+ 1
for even n, and (αn, βn, γn) = 43n2+ 14,34n2+32n−14,43n2+ 32n+34
for oddn. This entails, via algebraic cubatureand hyperinterpolation(discretized expansion in series of orthogonal polynomials) with respect to the product Chebyshev measure, that
Proposition 8 The rank-1 Chebyshev cubature lattice An = {`n(sπ/ν)}, s = 0, . . . , ν =
nγn+ 1, is a WAM for the cube, withC(An) =O(log3n) and card(An)∼ 34n3.
The interest for trivariate function approximation by sampling along Lissajous curves arises, for example, in the emerging field of MPI (Magnetic Particle Imaging, cf. [12] and the references therein). The construction of 3d Lissajous WAMs is studied in [4], and is implemented by the Matlab functionwamlissa of [10].
2
Interpolation and fitting
We recall here some basic results and algorithms concerning polynomial fitting on WAMs, and polynomial interpolation at discrete extremal sets extracted from WAMs. Let us term LAn the projection operator C(K) → P
d
n defined by polynomial least squares on a WAM,
and LFn the projection operator defined by interpolation on Fekete points of degree n,
say Fn,extracted from a WAM (these are points that maximize the absolute value of the Vandermonde determinant). Concerning their operator norms with respect to k · kK,
kLAnk.C(An) p card(An), kLFnk ≤N C(An), N =Nn= dim(P d n) = n+d d , (6) for polynomial least squares and for polynomial interpolation, respectively, which show that WAMs with slowly increasing constants C(An) and cardinalities are relevant structures for
multivariate polynomial approximation, cf. [6]. In practice, however, (6) turn out to be large overestimates of the actual operator norm growth (see Figure 1). A standard calculation for projection operators provides the estimate kf − LfkK ≤ (1 +kLk) infp∈
Pdnkf−pkK,
∀f ∈ C(K), from which together with (6) we get convergence, whenever K is a Jackson compact and f a sufficiently regular function (cf. [21]).
2.1 Discrete Orthogonal Polynomials and Least Squares
The approximation algorithms start from Vandermonde-like matrices in suitable total-degree polynomial bases. The choice of the standard monomial basis is unappropriate already at small degrees, due to its severe ill-conditioning. A general and more suitable choice is the product Chebyshev basis of the smallest Cartesian rectangle containing the domain (say ×d s=1[as, bs] ⊇ K, d = 2,3), namely T(k1,...,kd)(x) = Qd s=1Tks 2xs−bs−as bs−as , 0 ≤Pd
s=1ks ≤ n, where Tk(·) = cos(karccos(·)) are the classical Chebyshev polynomials
of the first kind. By a suitable ordering (in [10] we adopted the graded lexicographical or-dering), we obtain a polynomial basis that we callp(x) = (p1(x), . . . , pN(x)), and we can
compute the corresponding Vandermonde-like matrix on a WAM of K
V(An,p) = (pj(ξi)) , 1≤i≤M , 1≤j≤N , (7)
where An = {ξ1, . . . ,ξM}. Notice that M ≥ N and V(An,p) is full-rank by (1), cf. [6].
The core of the fitting and interpolation procedures is a two-step discrete orthogonalization of the polynomial basis by the QR algorithm, namely V(An,p) =Q1R1,Q1=QR2, where
Q=V(An,ϕ) =V(An,p)R1−1R−21 (8)
is the (numerically) orthogonal Vandermonde-like matrix corresponding to the discrete or-thonormal polynomial basis ϕ = (ϕ1, . . . , ϕN) = (p1, . . . , pN)R−11R2−1; cf. [5, 24]. The
reason for iterating the QR factorization is to cope with the strong ill-conditioning, which is typical of Vandermonde-like matrices and increases with the degree. Two orthogonalization iterations generally suffice, unless the original matrixV(An,p) is so severely ill-conditioned (rule of thumb: condition number greater than the reciprocal of machine precision) that the algorithm may fail. In practice, the change of polynomial basis is conveniently implemented by the Matlab matrix right division operator [16], as ϕ = (p/R1)/R2, in view of the
ill-conditioning inherited by the triangular matricesR1andR2. We can compute least squares
polynomial projection of f ∈ C(K) on an array of target points X = {x1, . . . ,xS} ⊂ K,
and estimate the norm of the least squares projection operator, that we call its “Lebesgue constant” by analogy with interpolation, on an array of control pointsY, as
LAnf(X) =V(X,ϕ)Q
tf , f = (f(x
1), . . . , f(xS))t; kLAnk ≈ kQ(V(Y,ϕ))
tk
1 , (9)
cf. [5]. In the package [10], computation of Discrete Orthogonal Polynomialsϕ and evalu-ation of V(X,ϕ) are implemented by the Matlab functions wamdopand wamdopeval, least squares fitting by wamfit, and estimation of the Lebesgue constant bywamleb.
2.1.1 Mesh compression
In many situations, WAMs can be very large sets and thus the fitting procedure becomes computationally heavy. This happens, for example, already in 2d with many-sided polygons (cf. Proposition 2), or with Admissible Meshes of smooth convex domains (cf. Proposition 5). Moreover, when the sampling process is difficult or costly, it could be convenient to reduce in any case the WAM cardinality, resorting to the following “compression” result.
Proposition 9 Let An be a WAM of a compact set K ⊂ Rd with card(An) > N2n =
dim(P2n). Then there exist a WAM A∗n ⊂ An with card(A∗n) ≤ N2n and C(A∗n) =
C(An)
p
card(An).
The proof of Proposition 9 rests on a generalized version of the well-knownTchakaloff ’s theorem(cf. [23]), on the existence of low cardinality algebraic cubature formulas for com-pactly supported measures, in particular fordiscrete measures(we consider here the discrete measure with unit mass at each WAM point). Indeed, by such a theorem there exist a sub-setA∗
n={ξi1, . . . ,ξiV} ⊂ An,V ≤N2n, and positive weights w={wi1, . . . , wiV}, such that PM
i=1q(ξi) =
PV
k=1wikq(ξik) for everyq ∈P2n. Then for every p∈Pn, with q=p
2 we get kpkK C(An) ≤ kpkAn ≤ kpk`2(A n)=kpk`2w(A∗n) ≤ p kwk1kpkA∗ n = p card(An)kpkA∗ n.
The computation of the nodes{ξik}and weights{wik}can be formulated as the problem
of finding asparse(nonnegative) solution to theunderdetermined Vandermonde-like linear system (consider column vectors)
cf. (7), wherem= (m1, . . . , mN)tis the vector of discrete moments of the polynomial basis.
Sparsity can be achieved by reformulating (10) asNonNegative Least Squares (NNLS) problem minu≥0kVtu−mk2, and solving it by the Matlab function lsqnonneg which
uses a variant of the active set method by Lawson and Hanson (cf. [16]). Alternatively, since lsqnonneg can be very slow on large problems, we can solve (10) by QR with col-umn pivoting, implemented by the Matlab mldivide (or backslash) operator. The latter approach does not guarantee positivity, but it turns out in practice that in the consid-ered degree ranges the negative weights are few and small, hence the stability parameter
ρn = PVk=1|wik|/|
PV
k=1wik| is not far from 1, and C(A
∗
n) = C(An)
p
ρncard(An) has a
size comparable to that appearing in Proposition 9. Both methods are implemented in the Matlab function wamcomprexof [10], where one preliminary orthogonalization step is made to pull down the conditioning of V (see Figure 1 for an example of compression).
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 16 18 20 1 2 3 4 5 6 7 8 9 10
Figure 1: Left: WAM for degree n = 15 (·, about 2800 points) by triangulation on a polygon with 14 sides, and its compression into 496 = dim(P22n) points (◦); Right: Lebesgue
constants of the LS operator on the original (∗) and the compressed WAM (◦),n= 1, . . . ,20.
2.2 Discrete Extremal Sets
The search for good sets for multivariate polynomial interpolation has received renewed at-tention in recent years; cf., e.g., [3, 14, 17, 27]. We consider here the approximate versions of Fekete points of K (points that maximize the absolute value of the Vandermonde deter-minant) studied in [2, 3, 24]. Indeed, the continuum Fekete points are explicitly known only in two univariate instances (interval and complex circle), and are very difficult to compute. By (6), it makes sense to start from a WAM, that is from the corresponding orthogonal Vandermonde-like matrixQ=V(An,ϕ) in (8) (which is preferable for conditioning issues).
The problem of selecting aN×N square submatrix with maximal determinant from a given
M ×N rectangular matrix is known to be NP-hard, but can be solved in an approximate way by two simple greedy algorithms, that are fully described and analyzed in [3]. These
algorithms produce two interpolation nodal sets, calledDiscrete Extremal Sets.
The first algorithm, that computes the so-called Approximate Fekete Points (AFP), tries to maximize iteratively submatrix volumes until a maximal volumeN ×N submatrix of Q is obtained, and can be based on QR factorization with column pivoting, applied to
Qt (that in Matlab is implemented by the mldivide or backslash operator, cf. [16]). The notion of volume generated by a set of vectors generalizes the geometric concept related to parallelograms and parallelepipeds (the volume and determinant notions coincide on a square matrix). The second algorithm, that computes the so-called Discrete Leja Points (DLP), tries to maximize iteratively submatrix determinants, and is based simply on Gaus-sian elimination with row pivoting applied to Q. Denoting by A the M×2 matrix of the WAM nodal coordinates, the corresponding computational steps, in a Matlab-like style, are
w=Q\v; i=find(w6=0); FnAF P =A(i,:); (11)
for AFP, wherev is any nonzero N-dimensional vector, and
[L, U,π] =LU(Q,“vector”); i=π(1 :N); FnDLP =A(i,:); (12)
for DLP. In (12), we refer to the Matlab version of the LU factorization that produces a row permutation vector π. In both algorithms, we eventually select an index subset
i = (i1, . . . , iN), that extracts an approximate discrete extremal set Fn of the region K
from the WAM An. Algorithms (11) and (12) are implementd by the Matlab function
dexsetsof [10]. Once one of the discrete extremal sets has been computed, we can simply apply (8)-(9) with theN×N matrixV(Fn,p) substituting V(An,p), in order to compute
the interpolation polynomial LFn and to estimate the Lebesgue constant kLFnk, by the
Matlab functions wamfit and wamleb in [10]. The two families of discrete extremal sets share the same asymptotic behavior which, by a recent deep result in pluripotential theory, is exactly that of the discrete uniform probability measures associated with the continuum Fekete points; cf. [1, 2, 3] and the references therein.
Proposition 10 Let Fn = {ξi1, . . . ,ξiN} the AFP or DLP extracted from a WAM of a
compact set K ⊂Rd (or K ⊂
Cd). Then limn→∞ N1 PNk=1f(ξik) =
R
Kf(x)dµK for every
f ∈C(K), where µK is the equilibrium measure of K.
Moreover, in all our numerical experiments both AFP and DLP have shown good computational features, with a Lebesgue constant growing more slowly thanN C(An), the theoretical upper bound in (6) for Fekete points extracted from a WAM (usually with a better behavior of AFP with respect to DLP); cf., e.g., [2, 3, 4, 9, 11, 25]. See Figures 2 and 3 for some examples of Discrete Extremal Sets on different 2d and 3d geometries.
It is also worth recalling that (differently from AFP) DLP are asequence, i.e., the first
Pds. Exploiting this feature, in [10] we provide a file of precomputed DLP for trivariate
polynomial interpolation up to degreen= 30, on the Lissajous curve of the cube described in Subsection 1.2.2 (by affine transformation these points can be used in any parallelepiped).
−0.5 0 0.5 1 1.5 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 −1.5 −1 −0.5 0 0.5 1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8
Figure 2: 66 Approximate Fekete Points (◦) and Discrete Leja Points (∗) extracted from a WAM (·) for degree n= 10 on a symmetric lens (left) and on a lune (right).
−1 0 1 −1 −0.5 0 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
Figure 3: 56 Approximate Fekete Points (◦) for degree n= 5 extracted from a WAM (∗) on a solid torus section and on the Lissajous curve of the cube.
Acknowledgements
This work has been partially supported by the ex-60% funds and by the biennial projects CPDA124755/CPDA143275 of the University of Padova, and by the INdAM GNCS.
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